Fractal energy gaps and topological invariants in hBN/Graphene/hBN double moir\'{e} systems

TL;DR

This paper investigates the electronic structure of double moiré systems in graphene/hBN heterostructures, revealing fractal energy gaps and topological invariants related to quasi Brillouin zones, with potential implications for topological materials.

Contribution

It introduces a method to identify topological invariants of energy gaps in quasiperiodic double-moiré systems and characterizes these gaps using geometric areas in momentum space.

Findings

Energy spectrum contains multiple minigaps with fractal Hofstadter butterfly structure.

Energy gaps are characterized by integers linked to areas in momentum space.

Quasi Brillouin zones are polygons defined by Bragg planes, providing a geometric interpretation.

Abstract

We calculate the electronic structure in quasiperiodic double-moir\'e systems of graphene sandwiched by hexagonal boron nitride, and identify the topological invariants of energy gaps. We find that the electronic spectrum contains a number of minigaps, and they exhibit a recursive fractal structure similar to the Hofstadter butterfly when plotted against the twist angle. Each of the energy gaps can be characterized by a set of integers, which are associated with an area in the momentum space. The corresponding area is geometrically interpreted as a quasi Brillouin zone, which is a polygon enclosed by multiple Bragg planes of the composite periods and can be uniquely specified by the plain wave projection in the weak potential limit.